1. **State the problem:** Solve the quadratic equation $$-2t^2 - 10t + 5 = 0$$ by completing the square and applying the square root property. 2. **Rewrite the equation:** First, divide the entire equation by $$-2$$ to make the coefficient of $$t^2$$ equal to 1. $$\cancel{-2}t^2 - \cancel{10}t + \cancel{5} = 0 \implies t^2 + 5t - \frac{5}{2} = 0$$ 3. **Isolate the constant term:** Move the constant term to the right side. $$t^2 + 5t = \frac{5}{2}$$ 4. **Complete the square:** Take half of the coefficient of $$t$$, which is $$\frac{5}{2}$$, and square it. $$\left(\frac{5}{2}\right)^2 = \frac{25}{4}$$ Add this to both sides to keep the equation balanced. $$t^2 + 5t + \frac{25}{4} = \frac{5}{2} + \frac{25}{4}$$ 5. **Simplify the right side:** Convert $$\frac{5}{2}$$ to $$\frac{10}{4}$$ to add easily. $$\frac{10}{4} + \frac{25}{4} = \frac{35}{4}$$ 6. **Write the left side as a perfect square:** $$\left(t + \frac{5}{2}\right)^2 = \frac{35}{4}$$ 7. **Apply the square root property:** Take the square root of both sides. $$t + \frac{5}{2} = \pm \sqrt{\frac{35}{4}} = \pm \frac{\sqrt{35}}{2}$$ 8. **Solve for $$t$$:** $$t = -\frac{5}{2} \pm \frac{\sqrt{35}}{2} = \frac{-5 \pm \sqrt{35}}{2}$$ **Final answer:** $$t = \frac{-5 + \sqrt{35}}{2}$$ or $$t = \frac{-5 - \sqrt{35}}{2}$$